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Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing semirigid rods from a L×L square lattice that contains two layers (and M=L×L×2 sites). The process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by removing groups of k consecutive monomers according to a generalized random sequential adsorption mechanism. The study is conducted by following the behavior of two critical concentrations with size k: (1) jamming coverage θ_{j,k}, which represents the concentration of occupied sites at which the jamming state is reached, and (2) inverse percolation threshold θ_{c,k}, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. The obtained results indicate that (1) the jamming coverage exhibits an increasing dependence on the size k-it rapidly increases for small values of k and asymptotically converges towards a definite value for infinitely large k sizes θ_{j,kâ∞}≈0.2701-and (2) the inverse percolation threshold is a decreasing function of k in the range 1≤k≤17. For k≥18, all jammed configurations are percolating states (the lattice remains connected even when the highest allowed concentration of removed sites is reached) and, consequently, there is no nonpercolating phase. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid k-mers are randomly and irreversibly deposited on a square lattice forming two layers. In this case, percolating and nonpercolating phases extend to infinity in the space of the parameter k and the model presents percolation transition for the whole range of k. The results obtained in the present study were also compared with those reported for the case of inverse percolation by removal of rigid linear k-mers from a square monolayer. The differences observed between monolayer and bilayer problems were discussed in terms of vulnerability and network robustness. Finally, the accurate determination of the critical exponents ν, ß, and γ reveals that the percolation phase transition involved in the system has the same universality class as the standard percolation problem.
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Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (particles occupying k consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θ_{c,k}[Ï_{c,k}], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k}[Ï_{c,k}] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θ_{c,k}^{i}[Ï_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[Ï_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, Ï_{c,k}, θ_{c,k}^{i}, and Ï_{c,k}^{i} are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θ_{c,k}[θ_{c,k}^{i}] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θ_{c,kâ∞}=0.500(2) [θ_{c,kâ∞}^{i}=0.500(1)]; (2)Ï_{c,k}[Ï_{c,k}^{i}] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers Ï_{c,kâ∞}=0.5334(6) [Ï_{c,kâ∞}^{i}=0.4666(6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(Ï)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and Ï_{c,k}+Ï_{c,k}^{i}=1) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the θ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.
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Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal, and random lattices is studied. The adsorption process is modeled by using random sequential adsorption algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension d between 1 and 2, and on Erdos-Rényi random graphs. The number of sites is M=L^{d} for Euclidean and fractal lattices, where L is a characteristic length of the system. In the case of random graphs, such a characteristic length does not exist, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. This paper concentrates on measuring (i) the probability W_{L(M)}(θ) that a lattice composed of L^{d}(M) elements reaches a coverage θ and (ii) the exponent ν_{j} characterizing the so-called jamming transition. The results obtained for Euclidean, fractal, and random lattices indicate that the quantities derived from the jamming probability W_{L(M)}(θ), such as (dW_{L}/dθ)_{max} and the inverse of the standard deviation Δ_{L}, behave asymptotically as M^{1/2}. In the case of Euclidean and fractal lattices, where L and d can be defined, the asymptotic behavior can be written as M^{1/2}=L^{d/2}=L^{1/ν_{j}}, with ν_{j}=2/d.
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Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of k×k square tiles (k^{2}-mers) from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by occupied sites. Then the system is diluted by removing k^{2}-mers of occupied sites from the lattice following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be removed due to the absence of occupied sites clusters of appropriate size and shape. The central idea of this paper is based on finding the maximum concentration of occupied sites, p_{c,k}, for which the connectivity disappears. This particular value of the concentration is called the inverse percolation threshold and determines a well-defined geometrical phase transition in the system. The results obtained for p_{c,k} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤4. For k≥5, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the size k considered. In addition, the accurate determination of the critical exponents ν, ß, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and four, has the same universality class as the standard percolation problem.
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Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond k-mers (sets of k linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage p_{j,k} and percolation threshold p_{c,k} are determined for a wide range of k (1≤k≤120). p_{j,k} and p_{c,k} exhibit a decreasing behavior with increasing k, p_{j,kâ∞}=0.7476(1) and p_{c,kâ∞}=0.0033(9) being the limit values for large k-mer sizes. p_{j,k} is always greater than p_{c,k}, and consequently, the percolation phase transition occurs for all values of k. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond k-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold p_{c,k}^{i}, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage p_{j,k}^{i} is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that p_{j,k}^{i}=1-p_{j,k}. The obtained results for p_{c,k}^{i} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤18. For k>18, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds p_{j,k}^{i} is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes (k=1) are much more effective than correlated attacks on groups of close nodes (large k's). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size k considered.
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The influence of adding different polysaccharides (locust bean gum, LBG; methyl cellulose, MC; and carboxymethyl cellulose, CMC) to gluten-based biodegradable polymeric materials was assessed in this work. Gluten/polysaccharide/plasticiser bioplastics were prepared at different polysaccharide concentrations (0-4.5%) and pH values by mixing in a two-blade counter-rotating batch mixer (at 25 °C under adiabatic conditions) and thermomoulding at 9MPa and 130 °C. Bioplastic probes were evaluated through dynamic mechanical thermal analysis, tensile strength and water absorption capacity tests. Results pointed out that a moderate enhancement of the network structure may be achieved by adding polysaccharide at a pH close to the protein isoelectric point (pH 6), which also conferred a further thermosetting capacity to the system. Moreover, the addition of MC and CMC was found to significantly enhance material elongation properties. However, the presence of charges induced by pH leaded to a higher incompatibility between the polysaccharide and protein domains forming the composite. The pH value played a relevant role in the material water absorption, which significantly increased under acidic or basic conditions (particularly at pH 3).